CycleInverseClassSize

2 to the power of the cycles of length at least three: the permutations got by inverting any of p's cycles.

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CycleInverseClassSize(p)2 to the power of the cycles of length at least three: the permutations got by inverting any of p's cycles.

Typed signatures

Type	Package	Applies
(permutation) -> number	combinatorics	always

Domain: Permutation statistics

Details

• Defined over Permutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a Permutation and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

1 if Length(_x) < 1 else 2 ^ Count(Filter(Map(
                                               Function(Min(Filter(
                                                                    Range(
                                                                           1,
                                                                           Length(_x),
                                                                           1,
                                                                         ),
                                                                    Function(Fold(
                                                                                   Function(_x[w], w, j),
                                                                                   i,
                                                                                   Range(1, k, 1),
                                                                                 ) ==
                                                                             i, k),
                                                                  )), i),
                                               Filter(
                                                       Range(1, Length(_x), 1),
                                                       Function(Fold(
                                                                      Function(
                                                                                w if w <= 0 else 0 if w == i else -1 if w < i else _x[w],
                                                                                w,
                                                                                k,
                                                                              ),
                                                                      _x[i],
                                                                      Range(
                                                                             1,
                                                                             Length(_x),
                                                                             1,
                                                                           ),
                                                                    ) == 0, i),
                                                     ),
                                             ), Function(c >= 3, c)))

Examples

Basic #

the one-line word $312$

live ↗

In$\mathrm{CycleInverseClassSize}({\operatorname{Permutation}}([3, 1, 2]))$Out$2$

the one-line word $2413$

live ↗

In$\mathrm{CycleInverseClassSize}({\operatorname{Permutation}}([2, 4, 1, 3]))$Out$2$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1]), \text{CycleInverseClassSize})$Out$1$

Known value · FindStat St001359

FindStat's value for [1,2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{CycleInverseClassSize})$Out$1$

Known value · FindStat St001359

FindStat's value for [2,1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{CycleInverseClassSize})$Out$1$

Known value · FindStat St001359

About

Examples	5
Crosswalk	FindStat